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Groups Complexity Cryptology

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Editorial Board Member: Blackburn, Simon R. / Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / May, Alexander / Mikhalev, Alexander V. / Myasnikov, Alexei / Roman'kov, Vitalii / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura / Weil, Pascal

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Mathematical Citation Quotient 2013: 0.12

Random equations in free groups

1Department of Mathematical Sciences, Stevens Institute of Technology, New Jersey, USA.

2Institute of Mathematics and Information Technologies, Omsk State Dostoevskii University, Russia.

Citation Information: Groups – Complexity – Cryptology. Volume 3, Issue 2, Pages 257–284, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: 10.1515/gcc.2011.010, November 2011

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In this paper we study the asymptotic probability that a random equation in a finitely generated free group F is solvable in F. For one-variable equations this probability is zero, but for split equations, i.e., equations of the form v(x 1, . . . , xk) = g, gF, the probability is strictly between zero and one if k ≥ rank(F) ≥ 2. As a consequence the endomorphism problem in F has intermediate asymptotic density, and we obtain the first natural algebraic examples of subsets of intermediate density in free groups of rank larger than two.

Keywords.: Free abelian groups; free groups; random equations; asymptotic density

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